American Institute of Mathematical Sciences

Advanced
April  2014, 34(4): 1375-1396. doi: 10.3934/dcds.2014.34.1375

On the Lagrangian structure of quantum fluid models

Philipp Fuchs 1, , Ansgar Jüngel 1, and  Max von Renesse 2,

1. 

Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraße 8-10, 1040 Wien, Austria, Austria
2. 

Institute for Mathematics, Universität Leipzig, Augustusplatz 10, 04109 Leipzig, Germany

Received  January 2013 Revised  May 2013 Published  October 2013

Some quantum fluid models are written as the Lagrangian flow of mass distributions and their geometric properties are explored. The first model includes magnetic effects and leads, via the Madelung transform, to the electromagnetic Schrödinger equation in the Madelung representation. It is shown that the Madelung transform is a symplectic map between Hamiltonian systems. The second model is obtained from the Euler-Lagrange equations with friction induced from a quadratic dissipative potential. This model corresponds to the quantum Navier-Stokes equations with density-dependent viscosity. The fact that this model possesses two different energy-dissipation identities is explained by the definition of the Noether currents.
Keywords: magnetic Schrödinger equation, symplectic form, Noether theory, quantum Navier-Stokes equations, osmotic velocity., Geometric mechanics.
Mathematics Subject Classification: Primary: 35Q40, 37J10, 58D3.
Citation: Philipp Fuchs, Ansgar Jüngel, Max von Renesse. On the Lagrangian structure of quantum fluid models. Discrete & Continuous Dynamical Systems, 2014, 34 (4) : 1375-1396. doi: 10.3934/dcds.2014.34.1375
References:
[1]

L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in the Space of Probability Measures,'' Second edition, Lectures in Mathematics ETH Zürich, Birkhäuser, Basel, 2008.  Google Scholar

[2]

A. Arnold, Mathematical properties of quantum evolution equations, in "Quantum Transport Modelling, Analysis and Asymptotics'' (eds. G. Allaire, A. Arnold, P. Degond and T. Y. Hou), Lecture Notes in Mathematics, 1946, Springer, Berlin, (2008), 45-109. doi: 10.1007/978-3-540-79574-2_2.  Google Scholar

[3]

V. Arnold, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits, Ann. Inst. Fourier (Grenoble), 16 (1966), 319-361. doi: 10.5802/aif.233.  Google Scholar

[4]

J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math., 84 (2000), 375-393. doi: 10.1007/s002110050002.  Google Scholar

[5]

S. Brull and F. Méhats, Derivation of viscous correction terms for the isothermal quantum Euler model, Z. Angew. Math. Mech., 90 (2010), 219-230. doi: 10.1002/zamm.200900297.  Google Scholar

[6]

B. van Brunt, "The Calculus of Variations,'' Universitext, Springer-Verlag, New York, 2004.  Google Scholar

[7]

P. Dirac, The Lagrangian in quantum mechanics, Phys. Z. Sowjet., 3 (1933), 64-72. Google Scholar

[8]

Dj. Djukic and B. Vujanović, Noether's theory in classical nonconservative mechanics, Acta Mech., 23 (1975), 17-27.  Google Scholar

[9]

E. Feireisl, "Dynamics of Viscous Compressible Fluids,'' Oxford Lecture Series in Mathematics and its Applications, 26, Oxford University Press, Oxford, 2004.  Google Scholar

[10]

J. Feng and T. Nguyen, Hamilton-Jacobi equations in space of measures associated with a system of conservation laws, J. Math. Pures Appl. (9), 97 (2012), 318-390. doi: 10.1016/j.matpur.2011.11.004.  Google Scholar

[11]

L. Brown, ed., "Feynman's Thesis. A New Approach to Quantum Theory,'' World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. doi: 10.1142/9789812567635.  Google Scholar

[12]

T. Frankel, "The Geometry of Physics. An Introduction,'' Cambridge University Press, Cambridge, 1997.  Google Scholar

[13]

G. Frederico and D. Torres, Nonconservative Noether's theorem in optimal control, Intern. J. Tomogr. Stat., 5 (2007), 109-114.  Google Scholar

[14]

J.-L. Fu and L.-Q. Chen, Non-Noether symmetries and conserved quantities of nonconservative dynamical systems, Phys. Lett. A, 317 (2003), 255-259. doi: 10.1016/j.physleta.2003.08.028.  Google Scholar

[15]

A. Jüngel, "Transport Equations for Semiconductors,'' Lecture Notes in Physics, 773, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-89526-8.  Google Scholar

[16]

A. Jüngel, Global weak solutions to compressible Navier-Stokes equations for quantum fluids, SIAM J. Math. Anal., 42 (2010), 1025-1045. doi: 10.1137/090776068.  Google Scholar

[17]

A. Jüngel, J. L. López and J. Montejo-Gámez, A new derivation of the quantum Navier-Stokes equations in the Wigner-Fokker-Planck approach, J. Stat. Phys., 145 (2011), 1661-1673. doi: 10.1007/s10955-011-0388-3.  Google Scholar

[18]

A. Jüngel and J.-P. Milišić, Full compressible Navier-Stokes equations for quantum fluids: Derivation and numerical solution, Kinetic Related Models, 4 (2011), 785-807. doi: 10.3934/krm.2011.4.785.  Google Scholar

[19]

J. Lafferty, The density manifold and configuration space quantization, Trans. Amer. Math. Soc., 305 (1988), 699-741. doi: 10.1090/S0002-9947-1988-0924776-9.  Google Scholar

[20]

J. Lott, Some geometric calculations on Wasserstein space, Commun. Math. Phys., 277 (2008), 423-437. doi: 10.1007/s00220-007-0367-3.  Google Scholar

[21]

E. Madelung, Quantentheorie in hydrodynamischer Form, Z. Phys., 40 (1926), 322-326. doi: 10.1007/BF01400372.  Google Scholar

[22]

P. Markowich, T. Paul and C. Sparber, Bohmian measures and their classical limit, J. Funct. Anal., 259 (2010), 1542-1576. doi: 10.1016/j.jfa.2010.05.013.  Google Scholar

[23]

R. McCann, Polar factorization of maps on Riemannian manifolds, GAFA Geom. Funct. Anal., 11 (2001), 589-608. doi: 10.1007/PL00001679.  Google Scholar

[24]

E. Nelson, Derivation of the Schrödinger equation from Newtonian mechanics, Phys. Rev., 150 (1966), 1079-1085. doi: 10.1103/PhysRev.150.1079.  Google Scholar

[25]

F. Otto, The geometry of dissipative evolution equations: The porous-medium equation, Commun. Part. Diff. Eqs., 26 (2001), 101-174. doi: 10.1081/PDE-100002243.  Google Scholar

[26]

F. Otto and C. Villani, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality, J. Funct. Anal., 173 (2000), 361-400. doi: 10.1006/jfan.1999.3557.  Google Scholar

[27]

M.-K. von Renesse, On optimal transport view on Schrödinger's equation, Canad. Math. Bull., 55 (2012), 858-869. doi: 10.4153/CMB-2011-121-9.  Google Scholar

[28]

W. Sarlett and F. Cantrijn, Generalization of Noether's Theorem in classical mechanics, SIAM Review, 23 (1981), 467-494. doi: 10.1137/1023098.  Google Scholar

[29]

R. Talman, "Geometric Mechanics,'' Wiley, New York, 2000. Google Scholar

show all references

References:
[1]

L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in the Space of Probability Measures,'' Second edition, Lectures in Mathematics ETH Zürich, Birkhäuser, Basel, 2008.  Google Scholar

[2]

A. Arnold, Mathematical properties of quantum evolution equations, in "Quantum Transport Modelling, Analysis and Asymptotics'' (eds. G. Allaire, A. Arnold, P. Degond and T. Y. Hou), Lecture Notes in Mathematics, 1946, Springer, Berlin, (2008), 45-109. doi: 10.1007/978-3-540-79574-2_2.  Google Scholar

[3]

V. Arnold, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits, Ann. Inst. Fourier (Grenoble), 16 (1966), 319-361. doi: 10.5802/aif.233.  Google Scholar

[4]

J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math., 84 (2000), 375-393. doi: 10.1007/s002110050002.  Google Scholar

[5]

S. Brull and F. Méhats, Derivation of viscous correction terms for the isothermal quantum Euler model, Z. Angew. Math. Mech., 90 (2010), 219-230. doi: 10.1002/zamm.200900297.  Google Scholar

[6]

B. van Brunt, "The Calculus of Variations,'' Universitext, Springer-Verlag, New York, 2004.  Google Scholar

[7]

P. Dirac, The Lagrangian in quantum mechanics, Phys. Z. Sowjet., 3 (1933), 64-72. Google Scholar

[8]

Dj. Djukic and B. Vujanović, Noether's theory in classical nonconservative mechanics, Acta Mech., 23 (1975), 17-27.  Google Scholar

[9]

E. Feireisl, "Dynamics of Viscous Compressible Fluids,'' Oxford Lecture Series in Mathematics and its Applications, 26, Oxford University Press, Oxford, 2004.  Google Scholar

[10]

J. Feng and T. Nguyen, Hamilton-Jacobi equations in space of measures associated with a system of conservation laws, J. Math. Pures Appl. (9), 97 (2012), 318-390. doi: 10.1016/j.matpur.2011.11.004.  Google Scholar

[11]

L. Brown, ed., "Feynman's Thesis. A New Approach to Quantum Theory,'' World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. doi: 10.1142/9789812567635.  Google Scholar

[12]

T. Frankel, "The Geometry of Physics. An Introduction,'' Cambridge University Press, Cambridge, 1997.  Google Scholar

[13]

G. Frederico and D. Torres, Nonconservative Noether's theorem in optimal control, Intern. J. Tomogr. Stat., 5 (2007), 109-114.  Google Scholar

[14]

J.-L. Fu and L.-Q. Chen, Non-Noether symmetries and conserved quantities of nonconservative dynamical systems, Phys. Lett. A, 317 (2003), 255-259. doi: 10.1016/j.physleta.2003.08.028.  Google Scholar

[15]

A. Jüngel, "Transport Equations for Semiconductors,'' Lecture Notes in Physics, 773, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-89526-8.  Google Scholar

[16]

A. Jüngel, Global weak solutions to compressible Navier-Stokes equations for quantum fluids, SIAM J. Math. Anal., 42 (2010), 1025-1045. doi: 10.1137/090776068.  Google Scholar

[17]

A. Jüngel, J. L. López and J. Montejo-Gámez, A new derivation of the quantum Navier-Stokes equations in the Wigner-Fokker-Planck approach, J. Stat. Phys., 145 (2011), 1661-1673. doi: 10.1007/s10955-011-0388-3.  Google Scholar

[18]

A. Jüngel and J.-P. Milišić, Full compressible Navier-Stokes equations for quantum fluids: Derivation and numerical solution, Kinetic Related Models, 4 (2011), 785-807. doi: 10.3934/krm.2011.4.785.  Google Scholar

[19]

J. Lafferty, The density manifold and configuration space quantization, Trans. Amer. Math. Soc., 305 (1988), 699-741. doi: 10.1090/S0002-9947-1988-0924776-9.  Google Scholar

[20]

J. Lott, Some geometric calculations on Wasserstein space, Commun. Math. Phys., 277 (2008), 423-437. doi: 10.1007/s00220-007-0367-3.  Google Scholar

[21]

E. Madelung, Quantentheorie in hydrodynamischer Form, Z. Phys., 40 (1926), 322-326. doi: 10.1007/BF01400372.  Google Scholar

[22]

P. Markowich, T. Paul and C. Sparber, Bohmian measures and their classical limit, J. Funct. Anal., 259 (2010), 1542-1576. doi: 10.1016/j.jfa.2010.05.013.  Google Scholar

[23]

R. McCann, Polar factorization of maps on Riemannian manifolds, GAFA Geom. Funct. Anal., 11 (2001), 589-608. doi: 10.1007/PL00001679.  Google Scholar

[24]

E. Nelson, Derivation of the Schrödinger equation from Newtonian mechanics, Phys. Rev., 150 (1966), 1079-1085. doi: 10.1103/PhysRev.150.1079.  Google Scholar

[25]

F. Otto, The geometry of dissipative evolution equations: The porous-medium equation, Commun. Part. Diff. Eqs., 26 (2001), 101-174. doi: 10.1081/PDE-100002243.  Google Scholar

[26]

F. Otto and C. Villani, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality, J. Funct. Anal., 173 (2000), 361-400. doi: 10.1006/jfan.1999.3557.  Google Scholar

[27]

M.-K. von Renesse, On optimal transport view on Schrödinger's equation, Canad. Math. Bull., 55 (2012), 858-869. doi: 10.4153/CMB-2011-121-9.  Google Scholar

[28]

W. Sarlett and F. Cantrijn, Generalization of Noether's Theorem in classical mechanics, SIAM Review, 23 (1981), 467-494. doi: 10.1137/1023098.  Google Scholar

[29]

R. Talman, "Geometric Mechanics,'' Wiley, New York, 2000. Google Scholar

[1]

C. Foias, M. S Jolly, I. Kukavica, E. S. Titi. The Lorenz equation as a metaphor for the Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2001, 7 (2) : 403-429. doi: 10.3934/dcds.2001.7.403
[2]

Chérif Amrouche, Nour El Houda Seloula. $L^p$-theory for the Navier-Stokes equations with pressure boundary conditions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1113-1137. doi: 10.3934/dcdss.2013.6.1113
[3]

Anna Amirdjanova, Jie Xiong. Large deviation principle for a stochastic navier-Stokes equation in its vorticity form for a two-dimensional incompressible flow. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 651-666. doi: 10.3934/dcdsb.2006.6.651
[4]

Maxim A. Olshanskii, Leo G. Rebholz, Abner J. Salgado. On well-posedness of a velocity-vorticity formulation of the stationary Navier-Stokes equations with no-slip boundary conditions. Discrete & Continuous Dynamical Systems, 2018, 38 (7) : 3459-3477. doi: 10.3934/dcds.2018148
[5]

Wendong Wang, Liqun Zhang, Zhifei Zhang. On the interior regularity criteria of the 3-D navier-stokes equations involving two velocity components. Discrete & Continuous Dynamical Systems, 2018, 38 (5) : 2609-2627. doi: 10.3934/dcds.2018110
[6]

Zujin Zhang. A Serrin-type regularity criterion for the Navier-Stokes equations via one velocity component. Communications on Pure & Applied Analysis, 2013, 12 (1) : 117-124. doi: 10.3934/cpaa.2013.12.117
[7]

Matthew Gardner, Adam Larios, Leo G. Rebholz, Duygu Vargun, Camille Zerfas. Continuous data assimilation applied to a velocity-vorticity formulation of the 2D Navier-Stokes equations. Electronic Research Archive, 2021, 29 (3) : 2223-2247. doi: 10.3934/era.2020113
[8]

Ansgar Jüngel, Josipa-Pina Milišić. Full compressible Navier-Stokes equations for quantum fluids: Derivation and numerical solution. Kinetic & Related Models, 2011, 4 (3) : 785-807. doi: 10.3934/krm.2011.4.785
[9]

Dario Bambusi, A. Carati, A. Ponno. The nonlinear Schrödinger equation as a resonant normal form. Discrete & Continuous Dynamical Systems - B, 2002, 2 (1) : 109-128. doi: 10.3934/dcdsb.2002.2.109
[10]

Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602
[11]

Jan W. Cholewa, Tomasz Dlotko. Fractional Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 2967-2988. doi: 10.3934/dcdsb.2017149
[12]

Silvia Cingolani, Mónica Clapp. Symmetric semiclassical states to a magnetic nonlinear Schrödinger equation via equivariant Morse theory. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1263-1281. doi: 10.3934/cpaa.2010.9.1263
[13]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276
[14]

P. Balseiro, M. de León, Juan Carlos Marrero, D. Martín de Diego. The ubiquity of the symplectic Hamiltonian equations in mechanics. Journal of Geometric Mechanics, 2009, 1 (1) : 1-34. doi: 10.3934/jgm.2009.1.1
[15]

Kuijie Li, Tohru Ozawa, Baoxiang Wang. Dynamical behavior for the solutions of the Navier-Stokes equation. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1511-1560. doi: 10.3934/cpaa.2018073
[16]

Hermenegildo Borges de Oliveira. Anisotropically diffused and damped Navier-Stokes equations. Conference Publications, 2015, 2015 (special) : 349-358. doi: 10.3934/proc.2015.0349
[17]

Hyukjin Kwean. Kwak transformation and Navier-Stokes equations. Communications on Pure & Applied Analysis, 2004, 3 (3) : 433-446. doi: 10.3934/cpaa.2004.3.433
[18]

Vittorino Pata. On the regularity of solutions to the Navier-Stokes equations. Communications on Pure & Applied Analysis, 2012, 11 (2) : 747-761. doi: 10.3934/cpaa.2012.11.747
[19]

Igor Kukavica. On regularity for the Navier-Stokes equations in Morrey spaces. Discrete & Continuous Dynamical Systems, 2010, 26 (4) : 1319-1328. doi: 10.3934/dcds.2010.26.1319
[20]

Igor Kukavica. On partial regularity for the Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2008, 21 (3) : 717-728. doi: 10.3934/dcds.2008.21.717

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (81)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy